A324851 Numbers > 1 divisible by the sum of their prime indices.

2, 4, 6, 12, 15, 16, 20, 30, 35, 36, 42, 48, 56, 88, 99, 112, 120, 126, 130, 135, 143, 144, 160, 162, 180, 192, 210, 216, 220, 221, 228, 231, 242, 250, 256, 270, 275, 280, 288, 297, 300, 308, 322, 330, 338, 360, 396, 400, 408, 429, 435, 440, 455, 468, 480, 493

Offset

1,1

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.

For any k>=2, let d be a divisor of k such that d > A056239(k). Then 2^(d-A056239(k))*k is in the sequence. Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - Robert Israel, Mar 19 2019

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   56: {1,1,1,4}
   88: {1,1,1,5}
   99: {2,2,5}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  130: {1,3,6}
  135: {2,2,2,3}

Maple

filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2], t=ifactors(n)[2]) = 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

Mathematica

Select[Range[2, 100], Divisible[#, Plus@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>PrimePi[p]*k]]&]

Prog

(PARI) isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k, 1])*f[k, 2])); } \\ Michel Marcus, Mar 19 2019

Crossrefs

Cf. A003963, A036844, A056239, A112798, A120383, A290822, A306844.

Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.

Sequence in context: A057830 A013916 A141113 * A348572 A334156 A255001

Adjacent sequences: A324848 A324849 A324850 * A324852 A324853 A324854

Keyword

nonn

Author

Gus Wiseman, Mar 18 2019

Status

approved